A TEST OF THE FORMATION MECHANISM OF THE BROAD LINE REGION IN ACTIVE GALACTIC NUCLEI

The predictions of the FRADO model can be done with the standard Shakura–Sunyaev disk model without any modifications. The disk model is specified by the value of the black-hole mass, M, and accretion rate, $\dot{M}$. The locally emitted flux, F, at a radius r, integrated over frequency, is then given by the conservation laws of the mass and angular momentum

$$\begin{eqnarray}&&F=\displaystyle \frac{3{GM}\dot{M}}{8\pi {r}^{3}}\left(1-\sqrt{\displaystyle \frac{3\,{R}_{\mathrm{Schw}}}{r}}\right),\end{eqnarray} \tag{ 1 }$$

and the local effective temperature of the disk is

$$\begin{eqnarray}&&F={\sigma }_{{\rm{B}}}{T}_{\mathrm{eff}}^{4},\end{eqnarray} \tag{ 2 }$$

where ${\sigma }_{{\rm{B}}}$ is the Stephan–Boltzman constant. We further assume that the disk radiates as a blackbody, since color-correction to the effective temperature is only important in the innermost parts of the disk (e.g., Czerny et al. 2011; Done et al. 2012).

In the case of the FRADO model, the inner radius of the BLR is set by the dust sublimation temperature, T_dust. This temperature, in general, is not known very precisely, and it depends on the chemical composition of the grain, and the size, as well as on the local pressure, and it is not well established. In general, silicates evaporate in the temperature as low as ∼800 K, while amorphous grain, and particularly graphite, can exists at much higher temperatures. The maximum temperature of the dust in the solar system spans the range of 1370–1770 K (Posch et al. 2007), and the measured dust temperature in the case of the active galaxy NGC 4151 varied with the nuclear flux, reaching up to 1500 K (Schnülle et al. 2013). In Czerny & Hryniewicz (2011), we used 1000 K as a reference, but Galianni & Horne (2013) in his study of NGC 5548 advocated 1670 K, with the difference due to different assumptions about the geometry. In the model below, T_dust is assumed to be a free parameter of the model.

Thus, in FRADO model, the inner radius of the BLR is set by the condition

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}(r)={T}_{\mathrm{dust}},\end{eqnarray} \tag{ 3 }$$

and can be calculated for an assumed T_dust as a function of the black-hole mass and accretion rate. Here we assume that the dust and gas are thermally coupled, which is justified in the upper disk layers and in the BLR clouds. The densities there are of the order of 10⁹ cm${}^{-3}$ or higher (see, e.g., Adhikari et al. 2016).

In observations, the delay is measured against the monochromatic flux (at 5100 Å, in the case of Hβ monitoring). This flux can be also calculated from the model from a given black-hole mass and accretion rate. In the long wavelength limit, where the frequency-dependent flux shows ${F}_{\nu }\propto {\nu }^{1/3}$ behavior this can be done analytically (see, e.g., Czerny & Hryniewicz 2011), including the proportionality coefficient. However, in this work, we calculate the monochromatic flux numerically,

$$\begin{eqnarray}&&{L}_{\nu }=\nu {\int }_{3\,{R}_{\mathrm{Schw}}}^{\infty }4\pi {B}_{\nu }({T}_{\mathrm{eff}}(r))2\pi {rdr},\end{eqnarray} \tag{ 4 }$$

so no additional approximations are involved. Here L_ν represents the frequency times luminosity, as customarily used in the RM, and we adopt the viewing angle of 60° since the observed luminosity is usually determined assuming the isotropic emission, corresponding to $i=60^\circ$.

Thus, for a fixed T_dust and the range of black-hole masses and accretion rates, we calculate both the inner radius of the BLR and the monochromatic flux L₅₁₀₀.

The applicability of the FRADO model as formulated above is based on several assumptions: the disk must be Keplerian, stationary, optically thick, and the total mass of the disk should not be much larger than the mass of the black hole. The first two assumptions give us the knowledge of the angular momentum distribution independently from the disk structure, so the viscously generated flux is then independent from the viscosity mechanism. The third assumption guarantees the independence of the emited radiation on the disk structure—emission is uniquely determined by the local flux. However, the break down of any of these assumptions means that the model predictions are not reliable. The issue of having a total mass that is too large could be solved since the models based on the self-consistent description of the gravitational field in the presence of the massive disk have been studied (e.g., Hure 2002). However, the required computations are complex, so we use a spherically symmetric potential and check the disk mass in exemplary solutions a posteriori.

The gravitational instability in the differentially rotating systems has been described by Safronov (1960) and Toomre (1964), and the criterion for the vertically averaged Keplerian disk is

$$\begin{eqnarray}&&Q=\displaystyle \frac{{c}_{s}{{\rm{\Omega }}}_{{\rm{K}}}}{\pi G{\rm{\Sigma }}}\lt 1,\end{eqnarray} \tag{ 5 }$$

where c_s is the sound speed, ${{\rm{\Omega }}}_{{\rm{K}}}$ is the local Keplerian angular velocity, and Σ is the disk surface density. We can also use the criterion locally, within the disk, for example, at the equatorial plane, and then this criterion can be expressed as

$$\begin{eqnarray}&&{Q}_{\mathrm{loc}}=\displaystyle \frac{M}{2\pi {r}^{3}{\rho }_{e}}\lt 1,\end{eqnarray} \tag{ 6 }$$

where ${\rho }_{e}$ is the local density in the disk at the equatorial plane.

In the outer parts of the disk, this criterion is easily satisfied (see Rice 2016 for a recent review). The current studies indicate that, in the accretion disks surrounding a supermassive black hole, four characteristic regions likely appear:

• 1.  
  (i) inner region, were self-gravity is negligible and the disk structure is well described by the standard Shakura–Sunyaev disk (Shakura & Sunyaev 1973);
• 2.  
  (ii) second region, where the self-gravity leads to the development of the marginally self-gravitating disk ($Q\sim 1$), with gravity working as a local viscosity law (Paczyński 1978; Laughlin & Rozyczka 1996; Collin & Hure 1999; Lodato & Rice 2004; Duschl & Britsch 2006; Rafikov 2009);
• 3.  
  (iii) third region, where the self-gravity effects become global and the instability is violent leading to super-sonic turbulence; this region exists only at some evolutionary stages of an active nucleus (Kawakatu & Wada 2008); and
• 4.  
  (iv) outer region, where self-gravity leads to star formation (Collin & Zahn 1999, 2008; Thompson et al. 2005; Wutschik et al. 2013).

In this paper, we concentrate on regions (i) and (ii), with particular attention payed to the transitions between (i) and (ii), and (ii) and (iii), as potentially related to the change in the disk structure leading to formation of the BLR.

In order to determine the transition radii, we need to know the disk structure, i.e., either the disk surface density or the density in the equatorial plane, depending on the criterion. Those quantities, unlike the disk effective temperature, are not set by the conservation laws for a stationary Keplerian disk but do depend on the assumptions about the viscosity, and on the details of the cooling efficiency. This last aspect, in turn, depends on the disk opacity, which varies with the disk optical depth. In the inner parts the disk opacity is dominated by the electron scattering. However, in the outer parts, other atomic processes play a role as well and they can affect both the disk surface density and the local density.

Thus, to determine the radius where the self-gravity effects start to be important, we need to determine the disk structure, taking into account the proper description of the local opacity. Early papers considered the vertically averaged disk structure (Paczyński 1978; Clarke 2009; Rafikov 2009; Rice & Armitage 2009). However, the predicted importance of the self-gravity depend considerably on the importance of the radiation pressure and the source of opacity. Simple criteria based on vertically averaged solutions from Shakura & Sunyaev (1973) imply different dependence of the transition radius on the black-hole mass and accretion rate (see the Appendix), and provide a very inaccurate approximation.

We thus perform calculations of the disk vertical structure taking into account all the relevant processes, basically following the work of Różańska et al. (1999). We neglect here the disk corona, but we introduce the self-gravity effects. Since we concentrate on the disk parts that are not too close to the black-hole horizon, we neglect the advection term, which, in general, should be also included (Abramowicz et al. 1988, Sa̧dowski et al. 2011).

2.2.1. Local Vertical Structure of the Accretion Disk in Local Self-gravity Region

The local vertical structure of the classical accretion disk is set by three equations: vertical hydrostatic equilibrium, vertical energy transfer (radiative and eventually convective), and vertical dissipation profile. The condition of the marginal self-gravity imposed a local condition of the density in the equatorial plane. However, as was shown in several papers (see Rice 2016 and the references therein), the dissipation in the marginally self-gravitating disk behaves, to some extent, as the modified α viscosity. Therefore, in both the inner, non-self-gravity region and in the outer region, we use the α prescription, but in the inner region α is set as an arbitrary parameter while in region (ii) α is determined by the condition that the local density in the equatorial plane satisfies the condition of the marginal self-gravity. Thus, effectively, α is constant in the innermost region, and rises in region (ii). The transition to region (iii) takes place when α reaches the value of one (Gammie 2001). Local equations are supplemented by the global conservation laws of the mass and angular momentum, and the radial pressure gradient are neglected so the circular motion velocity is determined by the local Keplerian orbits.

We thus solve three equations describing the disk vertical structure at each radius, r. They represent the hydrostatic equilibrium, the energy transfer (radiative and convective), and the dissipation profile. The hydrostatic equilibrium is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{\rho }\displaystyle \frac{{dP}}{{dz}}=-\displaystyle \frac{{GMz}}{{r}^{3}}-2\pi G{{\rm{\Sigma }}}_{z}\end{eqnarray} \tag{ 7 }$$

Equation (7) differs from the corresponding formula in Shakura & Sunyaev (1973) by the presence of the second term on the right hand side representing the local self-gravity force, compressing the disk and described as in Paczyński (1978). The quantity ${{\rm{\Sigma }}}_{z}$ is the surface density measured from the equatorial plane to the current value of the coordinate z, so it vanishes in the equatorial plane. We also include the effect of the disk mass being possibly comparable, or larger than the mass of the central black hole by simply adding the disk mass, within a given radius to the black-hole mass. So here the mass M includes both the mass of the black hole as well as the disk mass inside the radius r. A more careful description of the global gravity field, including the departure from spherical symmetry, is beyond the scope of the current paper. The pressure, P, includes both the gas pressure and the radiation pressure, and the gas pressure calculation in the partially ionized zone includes the proper computation of the mean molecular weight.

The energy transport in the vertical direction is described as

$$\begin{eqnarray}F & = & {F}_{\mathrm{rad}}=-\displaystyle \frac{16\sigma {T}^{3}}{3\kappa \rho },\quad \,\mathrm{if}\ \,{{\rm{\nabla }}}_{\mathrm{rad}}\leqslant {{\rm{\nabla }}}_{\mathrm{ad}},\\ F & = & {F}_{\mathrm{rad}}+{F}_{\mathrm{conv}},\quad \mathrm{if}\ \,{{\rm{\nabla }}}_{\mathrm{rad}}\gt {{\rm{\nabla }}}_{\mathrm{ad}}.\end{eqnarray} \tag{ 8 }$$

We thus allow for convection to transport a fraction of the energy dissipated in the disk. This convection is described in a standard way, as in the stellar interior. The opacity κ used here is the Rosseland mean, tabularized as a function of density and temperature. We use here the table from Seaton et al. (1994), obtained with atomic data from the Opacity Project, for log T > 4.0, and tables from Alexander et al. (1983) for log T < 3.8, and we interpolate between the two tables when necessary. Thus our opacity includes all the processes, such as electron scattering, free–free, atomic transitions for cosmically abundant elements (H, He, C, N, O, Ne, Na, Mg, Al, Si, S, A, Ca, and Fe, in numerous ionization states), including the line blanketing, as well as the dust and grain opacity. Therefore, this opacity table can be applied in the inner as well as in the outer disk regions. We assumed solar metallicity. We neglect the magnetic energy transfer, which, in principle, can also be important (e.g., Czerny et al. 2003; Begelman et al. 2015).

$$\begin{eqnarray}&&\displaystyle \frac{{dF}}{{dz}}=\displaystyle \frac{3}{2}\alpha P{{\rm{\Omega }}}_{{\rm{K}}},\end{eqnarray} \tag{ 9 }$$

where ${{\rm{\Omega }}}_{{\rm{K}}}$ is the local Keplerian angular velocity. Thus the energy production within the disk is described through the α parameter, as in the Shakura & Sunyaev (1973).

In the innermost region, the parameter α is set arbitrarily, in region (ii) the required value of α is calculated from the condition that the disk is marginally self-gravitating, i.e., α becomes a function of the radius r. Numerical simulations (Forgan et al. 2011) justify this approach to the description of region (ii). The transition to region (iii) happens when α becomes equal to one. The simple continuous disk description does not apply beyond this radius, where clumps form due to the efficient cooling.

2.2.2. Global Conservations Laws and Boundary Conditions for a Vertical Structure Computing

The integration of the disk vertical structure is performed from the top downward, to the disk equatorial plane. The standard conservation laws of mass and angular momentum form the the boundary conditions, as in the Shakura–Sunyaev disk. The total dissipated flux at the radius r is given by

$$\begin{eqnarray}&&{F}_{\mathrm{tot}}=\displaystyle \frac{3{GM}\dot{M}}{8\pi {r}^{3}}\left(1-\sqrt{\displaystyle \frac{3\,{R}_{\mathrm{Schw}}}{r}}\right),\end{eqnarray} \tag{ 10 }$$

and the effective temperature is thus given by

$$\begin{eqnarray}&&{T}_{\mathrm{eff}}={\left(\displaystyle \frac{{F}_{\mathrm{tot}}}{{\sigma }_{{\rm{B}}}}\right)}^{1/4},\end{eqnarray} \tag{ 11 }$$

where ${\sigma }_{{\rm{B}}}$ is the Stefan–Boltzman constant. Since we use the diffusion approximation for the energy transfer (see Equation (8)), the temperature at the outer edge of the disk atmosphere, T₀, is lower by a factor of ${2}^{1/4}$ than the effective temperature, as in the standard Eddington approximation (Mihalas 1978). The effective temperature is achieved at the optical depth $\tau =2/3$. The density at the disk surface is assumed to be 10⁻¹⁶ g cm⁻³ instead of 0 to allow for the opacity computation. The disk thickness is determined iteratively from the condition that the flux dissipated in the disk and described by Equation (8) reaches zero at the equatorial plane. The integration is performed by the second-order Runge–Kutta scheme with an adaptive step size.

In the region of the marginal self-gravity, an additional iteration is necessary, that is the adjustment of the viscosity parameter α, then dependent on the disk radius, to ensure that the disk density in the equatorial plane satisfies the condition of the marginal self-gravity ${Q}_{\mathrm{loc}}=1$ (see Equation (6)). Our approach is basically equivalent to the method used by Rafikov (2009), Clarke (2009), and Rice & Armitage (2009), but these authors did not consider the vertical structure of the disk.

In the case of the self-gravity model, we also calculate the monochromatic luminosity at 5100 Å using Equation (4).

Application of Equations (10) and (11) to the disk with self-gravity means that the prediction of the FRADO model for such a disk is unchanged by the local self-gravity effect; though, the internal disk structure may be strongly affected. The introduced assumption of the local blackbody emission has to be checked a posteriori since self-gravity may strongly reduce the disk surface density.

We use the time delay measurements of the Hβ emission line summarized in Du et al. (2015, 2016), which are compiled from the reverberation campaigns performed by different groups (Peterson et al. 1991, 1992, 1994, 1998, 1999, 2002, 2014; Dietrich et al. 1993, 1998, 2012; Stirpe et al. 1994; Korista et al. 1995; Santos-Lleó et al. 1997, 2001; Collier et al. 1998; Kaspi et al. 2000; Bentz et al. 2006, 2007, 2009b, ; Peterson 2014; Denney et al. 2006, 2010; Grier et al. 2012; Barth et al. 2013; Du et al. 2014, 2015, 2016; Pei et al. 2014; Wang et al. 2014, 2016). The time delays represent the emissivity-weighted radii of the BLR in those RM objects. The contributions of host galaxies to their 5100 Å luminosities have been subtracted mainly using high-resolution images from the Hubble Space Telescope (Bentz et al. 2009a, 2013; Du et al. 2014; Wang et al. 2014), or adopting the empirical relationship for several objects (see Du et al. 2015, 2016). The cosmological parameters used are the Hubble constant ${H}_{0}=67$ km s⁻¹ Mpc⁻¹, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.68$, and ${{\rm{\Omega }}}_{M}=0.32$.

We perform a set of computations for a range of black-hole masses and accretion rates with the aim to locate the radii where the effective temperature is equal to 800 and 1500 K, which we consider as a lower and upper limit for the dust sublimation temperature. The black-hole masses are taken from the range $3\times {10}^{5}\,{M}_{\odot }$ to ${10}^{10}\,{M}_{\odot }$. We use the dimensionless accretion rate

$$\begin{eqnarray}&&\dot{m}=\displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}},{\dot{M}}_{\mathrm{Edd}}=1.27\times {10}^{18}\left(\displaystyle \frac{M}{\,{M}_{\odot }}\right)\ [{\rm{g}}\,{{\rm{s}}}^{-1}].\end{eqnarray} \tag{ 12 }$$

and we do the computations for $\dot{m}=0.01$, 1.0, and 100. Quasars and bright AGNs are usually thought to be within the 0.01–1.0 range (e.g., Kelly et al. 2010; Suh et al. 2015). However, there are indications that a fraction of sources may be at highly super-Eddington accretion rates (Collin et al. 2002; Abolmasov & Shakura 2012; Du et al. 2014, 2015, 2016; Wang et al. 2014). The disk model we use is not well adjusted to highly super-Eddington accretion rates, so the results for the last value should be treated with much reservation, but we show them to indicate the possible trend.

The predicted location of the inner radius of the BLR within the frame of the FRADO model is given in Figure 1. The location depends strongly on the accretion rate $\dot{m}$ as well as on the adopted dust temperature.

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We now assume that the BLR size can be simply interpreted as the time delay, without any geometrical factors connected with the source inclination and the extension of the BLR. We also assume that the measured monochromatic luminosity represents the mean isotropic radiation flux. Under such assumptions, we obtain an expected delay versus the monochromatic source luminosity at 5100 Å, and we can compare it to the measured time delays in the sample of AGNs.

The result is shown in Figure 2. The picture looks different now, when the horizontal axis is replaced with the monochromatic luminosity. In this case, the dependence on the accretion rate practically disappears. There is a slight departure from the strait line in the case of large black-hole mass and low accretion rate since, in this case, the disk is colder, and the emission at 5100 Å is relatively close to the peak of the disk spectrum instead of following the classical Shakura–Sunyaev behavior at longer wavelengths, ${F}_{\nu }\propto {\nu }^{1/3}$. However, when no such departure is seen, the delay–luminosity relation follows the pattern ${\tau }_{\mathrm{del}}\propto {L}_{5100}$ exactly, as outlined in Czerny & Hryniewicz (2011). The predictions for the highest accretion rate, $\dot{m}$, are strongly biased: the position of the sublimation radius is well represented, but the monochromatic luminosity is overpredicted due to the negligence of the advection, important in the innermost part of the disk. This bias should be lower than a factor of 10 since the total luminosity of a slim disk saturates at $\sim 10\,{L}_{\mathrm{Edd}}$.

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The comparison of the model prediction with the set of the measuerd time delays looks favorable for the FRADO model (see Figure 2). Intermediate temperature between the adopted 800 and 1500 K seems appropriate. There are a few outliers but overall most of the sources follow the trend. The best fit is indeed obtained for the dust temperature ${T}_{\mathrm{dust}}=899.6$ K, not shown on the plot.

Obtained dust temperature depends on the cosmological model used to determine the observed L₅₁₀₀ luminosities. In general, it also depends on the viewing angle, i, here assumed to be 60° and the exact geometry of the BLR. The inclination enters into the disk flux through a $\cos (i)$ for a geometrically flat disk, as well as in the time delay if the cloud distribution has a 3D shape and is spatially extended. Here no such effects were introduced, and the simple relation between the inner BLR radius and the time delay was adopted. The reliable determination of the inclinations in radio-quiet AGNs still poses a considerable problem (see, e.g., Marin 2016; Middleton et al. 2016). If the typical inclination of the source is 30° instead of 60°, this will result is a systematic offset in the luminosity by a factor of $2\cos 30^\circ \sim 1.732$ provided by the model but the same offset will be included in the derivation of the luminosity from the data. This systematic shift of the model and the data shift would lead to a different value (1100 K) for the dust sublimation temperature, best representing the data. The observed dispersion around the model lines in Figure 2 is, in principle, due to the combination of the possible differences in the intrinsic geometry, viewing angles and dust sublimation temperature. The dispersion around the relation is only 0.25 dex, and the intrinsic scatter is only 0.21 (Du et al. 2016), which can be fully accounted for by the dispersion of the viewing angles, so additional intrinsic dispersion in the dust sublimation temperature seems unlikely.

The first effects of the self-gravity may appear when the Toomre parameter, given by Equation (6), becomes smaller than one. The position of this radius, calculated from the disk model without self-gravity effects included is shown in Figure 3. The transition radius between zone (i) and zone (ii) only weakly depends on the Eddington ratio at sub-Eddinton to Eddington luminosities, and rises with the black-hole mass, if measured in absolute units. The rise, however, is much slower than linear, so the value of the transition radius measured in Schwarzschild units drops with mass from 10⁴ for $5\times {10}^{5}\,{M}_{\odot }$ down to 10 for ${10}^{10}\,{M}_{\odot }$. When the accretion rate becomes much higher, the transition radius seems to level off for larger masses, and the whole disk becomes self-gravitating for ${10}^{10}\,{M}_{\odot }$. This thus happens below the absolute limit of $5\times {10}^{10}\,{M}_{\odot }$ for the mass of the accreting black hole derived by King (2016) at the basis of very approximate formulae for self-gravity effects.

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The difference between the local and global criterion (see Equations (6) and (5)) is small; usually the local instability close to the equatorial plane appears first, but it is followed by the whole disk instability at a distance no larger than a factor of two.

The radius, where the total disk mass inside that radius is equal to the black-hole mass, is larger than the transition between (i) and (ii) region in general. However, for the extreme case of a very large accretion rate ($\dot{m}=100$) the two radii are comparable (see dashed lines Figure 3). Our solution may not be accurate in the region where the disk mass inside the studied radius becomes larger than the black-hole mass. Here, a more careful approach, following for example Hure (2002), would provide a better approximation.

The dependence of the disk properties on the radius is not as smooth as in the case of the disk effective temperature in the FRADO model. The local disk structure depends on the opacity, which, in turn, has a complex dependence on the local density and temperature which vary both in the vertical and in the radial direction. The opacities used by us are reliable and show overall similarity to the opacities used by Thompson et al. (2005), as illustrated in Figure 4.

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The results depend on the adopted value of the viscosity coefficient in the innermost region. We assumed $\alpha =0.01$ as our reference value. A higher value of the viscosity parameter leads to a lower value of the disk surface density, the total disk mass decreases at a given radius, and the radius where the disk mass equals the black-hole mass moves outward (see Figure 5). A change in α from 0.01 to 0.1 moves the transition radius between the region (i) and (ii) by a factor of two for typical parameter values.

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From an observational point of view, it is more convenient to measure this radius in light days, and in reference to the disk monochromatic flux at 5100 Å, as is frequently used in the reverberation campaign (see Figure 6), as we did for the FRADO model. The time delay predicted by a transition between region (i) and region (ii) is, however, much shorter than it was previously, from 1 to 30 days even for high luminosity sources. This transition is thus not connected with the BLR formation, which happens further out.

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The second important radius is marked by the total disk mass becoming equal to the black-hole mass. This radius is located further out (see Figure 3). However, here the radius where the total mass equals the central mass have been computed without the self-gravity effects, i.e., not self-consistently since this radius is located beyond region (ii).

The transition between region (ii) and region (iii) happens further down, at larger radii. We thus calculated several disk models, with self-gravity included, up to the radius where $\alpha =1$, i.e., the self-gravitating disk stops cooling efficiently to prevent the fragmentation. Here the self-gravity effects are very important and they modify the disk structure considerably, in comparison with the standard model without the self-gravity effects.

In Figure 7, we show a comparison of the disk vertical structure close to the border between region (ii) and region (iii) for an exemplary black-hole mass and accretion rate, and we compare it to the standard disk structure calculated without any self-gravity effects. This specific solution has the temperature at the surface 1030 K (effective temperature 1225 K). The self-gravitating disk is then almost isothermal, with equatorial temperature only two times higher than the surface temperature, while the standard disk is much hotter inside and denser; the surface density is also higher, but the geometrical thickness of the disk is very similar in both solutions. The Rosseland mean opacity is very different in these two solutions since the standard disk is hot and fully ionized, with opacity provided by numerous bound-free and line transitions, while most of the self-gravitating disk is in the region of the opacity drop (see Figure 4). The total optical depth of the self-gravitating disk is more than three orders of magnitude lower, but the disk is not optically thin: the total optical depth is ∼300. Both solutions are gas-dominated apart from the uppermost region, where the density is very low.

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This decrease in the disk surface density in the self-gravitating solution in comparison with the standard disk is responsible for the slower rise in the total disk mass. We show that trend in Figure 5 marked with a green dashed line. The self-gravitating solution is not extended beyond the transition radius from regions (ii) to (iii) while the standard disk was formally extended beyond that region. The curve is only visible in the lower, expanded part of the plot  because, for the adopted parameters, the transition to the region (iii) happens at $\mathrm{log}R=16.42\,\mathrm{cm}$. There, the rise in the total mass is not yet considerable but the assumptions of the model used in this paper break down beyond that radius. This is true for all the solutions with accretion rates $\dot{m}\leqslant 1$: the transition to region (iii) happens (usually well) before the disk mass becomes comparable to the black-hole mass. For the most extreme case, $\dot{m}=100$, and for the black-hole mass of ${10}^{7}\,{M}_{\odot }$, or larger, the total mass becomes larger than the black-hole mass at smaller radii than the local self-gravity criteria (Equations (5) and (6)) are satisfied. Much more careful treatment of the disk accretion flow is certainly necessary in this parameter space.

An example of how the self-gravity affects the disk at various distances from the black hole is shown in Figure 8. Both disks do not practically differ in the inner region; though, two self-gravity effects were included at all radii: the rise of the central mass with the radius and the vertical component due to self-gravity. However, essential differences arise only after crossing the border between regions (i) and (ii), with the rise of the effective α. The disk thickness does not decrease significantly despite the drop in the equatorial pressure in comparison to the standard disk. This is the effect noticed by Thompson et al. (2005); though, their simplified description has lead to the disk height to the radius ratio of the order of 100, thus implying a narrow funnel formed by the dusty disk around the symmetry axis. Our method of solving for the disk vertical structure implies moderate and realistic disk height to the radius ratio of the order of a few percent; though, the disk atmosphere is already dust-dominated while the disk interior is in the opacity-dip zone.

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Figure 9 shows the expected delay if the BLR forms where the self-gravity becomes more violent, i.e., at the transition from region (ii) to region (iii). The delay is a smooth and monotonic function for higher accretion rates but, for the low accretion rate, the dependence on L₅₁₀₀ is more complex due to the opacity because with the decrease of the temperature the opacity first shows a rise at intermediate temperatures (this is sometimes called a metallicity bump), and then a rapid decrease, later inverted when dust and grains set in (see Figure 8). For higher accretion rates, the transition radius always happens at a radius with the effective temperature higher than $1000K$, i.e., in the region where there is no dust in the disk atmosphere. However, for $\dot{m}=0.01$, the relative position of the dust sublimation radius and the transition (ii) to (iii) radius depends on the black-hole mass. For low black-hole mass, the dust sets in at smaller radii, and the effective temperature at the transition radius is low. For example, for the black-hole mass $3\times {10}^{5}\,{M}_{\odot }$ the transition between regions (ii) and (iii) happens at $4.1\times {10}^{5}\,{R}_{\mathrm{Schw}}$, where the effective temperature is extremely low, only 64 K. This temperature rises with the black-hole mass to 285 K for ${10}^{7}\,{M}_{\odot }$, and to 1000 K at $3\times {10}^{7}\,{M}_{\odot }$.

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Since the transition between regions (ii) and (iii) takes place at a larger radius than the transition between (i) and (ii), we checked if the model is still self-consistent. The disk is always optically thick in the whole studied region. The disk mass does not rise too strongly in the marginal self-gravity solution (see Figure 8). The total mass of the disk up to the zone (iii) boundary, however, becomes quite large in the case of very large accretion rates and high masses. For $\dot{m}=1$, the disk mass reaches 50% of the black-hole mass in the case of a high-mass black hole $M=3\times {10}^{9}\,{M}_{\odot }$. This still does not pose much of a problem to the model: disk mass is included (using a spherically symmetric approximation), and the local departure from the Keplerian motion is at a level of 0.5%, or less. However, for $\dot{m}=100$ and $M=3\times {10}^{9}\,{M}_{\odot }$, the disk total mass exceeds the black-hole mass by a factor of 20, the relative correction to the Keplerian motion due to the radial pressure gradient is 70%, and the heat advection carries 25% of the energy. Thus, if such extreme accretion rates in quasars containing very massive black holes are confirmed, the model has to be refined following the use of the slim disk (Abramowicz et al. 1988) concept combined with a better description of the gravity (Hure 2002). Currently, most quasars are considered to be radiating between 0.01 and 1 in Eddington units (see, e.g., Kelly et al. 2010).

We can again compare the predicted delay, this time based on the assumption that the transition between regions (ii) and (iii) is responsible for the onset of the turbulence and subsequent formation of the BLR with the data sample. The data points are not consistent with expectations. The expected time delays are too short in comparison with the model for luminous objects. At the lower luminosity end, the agreement seems better. In that region, the sources having higher mass (lower $\dot{m}$) for the same L₅₁₀₀ should show longer time delays than smaller mass objects.